Posted
by
timothy
on Sunday June 05, 2011 @10:12AM
from the baseball-hailstones-also-a-tough-problem dept.

mikejuk writes "A proof [preprint PDF] has been proposed for the Collatz conjecture about hailstone sequences. A hailstone sequence starts from any positive integer n the next number in the sequence is n/2 if n is even and 3n+1 if n is odd. The conjecture is that this simple sequence always ends in one. Simple to state but very difficult to prove and it has taken more than 60 years to get close to a solution."

That's not a fault in the strict sense, that's an omission. Besides, this IS a preprint - it's not been published, hasn't even been refereed or peer-reviewed yet, so it's not in the least bit surprising it's not perfect.

Remember Wiles' proof of FLT? That one went through quite a few iterations, and it took years until all the flaws were ironed out - big ones, in some cases, that required significant changes to the whole thing. But in the end, the proof stood.

A hailstone sequence starts from any positive integer n the next number in the sequence is n/2 if n is even and 3n+1 if n is odd.

It wouldn't have taken any more time to properly punctuate this "sentence" once -- for everyone -- than it takes everyone to punctuate it in their heads in order to make sense of it. I realize they just cut and pasted the bulleted points -- minus the bullets -- but c'mon, they didn't put those there just for decoration.

The reason mathematicians are interested in the 3n+1 problem is that we do not have a very good understanding of the process -- it is a fairly simple process to describe, but it is not so easy to explain why it would always fall into the same cycle (1,4,2,1,4,2,1,4,2). A lot of problems in number theory are like this; Fermat's last theorem was similarly easy to state but very difficult to understand and prove. The real-life application of these problems is often more related to the various methods required to solve them than the problem itself.

For example, consider the problem of the quintic formula. As everyone with a middle school education should know, there is a formula that gives the solution to any linear equation (ax+b = 0), and there is a formula that gives the solutions to any quadratic equation (ax^2 + bx + c = 0). Some more educated people will also know that there are formulas for cubic equations (ax^3 + bx^2 + cx + d = 0) and quartic equations (ax^4 + bx^3 + cx^2 + dx + e = 0). The obvious question is, "What about quintic equations (ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0)?"

The answer is a somewhat intellectually interesting, "No, there is no formula that gives the solution to all quintic equations (using only arithmetic and radicals)." There is no real-world application of that answer; we can get good enough approximations of the solutions to quintic equations by various numeric methods, which is perfectly fine for any problem that involves solving a quintic. However, the proof that there is no quintic formula involves fields of mathematics that are very much applicable to real-world problems: group theory and field theory are very important in cryptography and certain branches of physics. Additionally, those fields of study led to the research of more general abstract algebra, with still more real-world application.

So, no, there is no real-world use for the Collatz conjecture, at least none that I am aware of. In all likelihood, the proof of the Collatz conjecture will lead to some practical application, or a better understanding of certain real world problems.